Critical constraint on inflationary magnetogenesis

Shuichiro Yokoyama; Tomohiro Fujita

REVIEW

Critical constraint on inflationary magnetogenesis

Tomohiro Fujita, Shuichiro Yokoyama

Cosmology and Nongalactic Astrophysics · Submitted 4 Feb 2014 · 56 references

Pith's verdict: The 30 GeV bound is real and checkable; the 'any n' extension is a plausible worst case rather than a proof, and that is the only place the paper overreaches.

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CitationShuichiro Yokoyama, Tomohiro Fujita. "Critical constraint on inflationary magnetogenesis." arXiv:1402.0596 (2014). https://pith.science/paper/YNQS22JZ

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Editor's note

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The thing to know first: the central bound is right. Requiring that the electric field that necessarily accompanies inflationary magnetogenesis not overproduce curvature perturbations gives rho_inf^{1/4} < 30 GeV for B_obs = 10^-15 G at p_B = 1 Mpc^-1. I re-derived the chain from (3.18) through (4.9) and the coefficient checks out. The logic is hard to escape: to beat adiabatic dilution you need A growing faster than a^2, which forces E/B ~ n e^{N_k} on superhorizon scales, which means a large electric field at the end of inflation, which sources zeta through the non-adiabatic pressure. That is the paper, and it is a good argument. The truncation to the last e-fold is genuinely conservative for constant n, since the integrand eta^{2-2n} grows toward earlier times for n > 2, so extending the window would only tighten the bound. What is new relative to refs. 35 and 38 is that they do not assume the inflaton sources the dominant curvature perturbation and do not fix the inflation model. That is a real generalization. The conservative choices are all in the right direction and stated: dropping B in the four-point, dropping anisotropic stress, taking the minimal Maxwell Lagrangian. Section 5's list of escape routes is honest. Soft spots, in order. The |n| << 1 extension (4.13-4.17) assumes a specific piecewise history: B proportional to a^-4 back to N_c, then a sharp power law with n ~ O(1) over one e-fold. That is a plausible worst case, not a derivation for arbitrary A(eta), and the 'for any n' in (4.18) claims more than the argument delivers. It does not touch the physically relevant regime, n > 2, where B actually grows and (4.12) applies. Minor: eq. (4.2) as printed reads n^2 k^2 eta^2 P_B, which is dimensionally wrong; it should be n^2/(k^2 eta^2) P_B, which is what (4.5) actually uses. A typo, not an error in the derivation, but it will stop a careful reader. And the whole thing rides on the blazar lower bound, uncertain by a few orders; the authors say so, but 30 GeV should be read as an order-of-magnitude statement. Who it is for: anyone working on magnetogenesis, and anyone who wants a clean worked example of a spectator sector's isocurvature feeding zeta. Worth a serious referee. The bound is a keeper; the 'any n' phrasing should be softened.

Referee report

Significance

If correct, the result is a strong, essentially model-independent constraint that closes off standard inflationary magnetogenesis on Mpc scales unless the magnetic field is amplified after inflation or the strong-coupling regime is invoked. The derivation is parameter-free given n, the n-dependence is bounded by f(n) ≤ 1 for |n| ≥ 1, and the paper makes a falsifiable prediction (r < 7×10^{-61} in single-field slow roll, so a future tensor-mode detection would exclude the scenario). I checked the arithmetic of eqs. (4.9)-(4.10) and reproduced the 30 GeV coefficient, and I checked the r estimate in §5; both are correct. The conservative choices (dropping B contributions and anisotropic stress, and using ε ≤ 1) are appropriate and are stated as such.

Major objections (1)

[Section 4, eqs. (4.13)-(4.18)] The claim that eq. (4.18) holds 'for any n' rests on the N_c argument, which assumes a specific piecewise history: A ∝ η^{-n} with n ~ O(1) over [N_c, N_c+1], and n ≈ 0 (so P_B ∝ a^{-4}) from N_c to the end of inflation. For a smooth power-law history this is a reasonable worst case, and I agree the constraint strengthens at earlier times because of the e^{4N_c} factor in eq. (4.16). But for genuinely non-power-law A(η) — resonant amplification, non-perturbative production, or the strong-coupling regime — the argument is not established, and §5 acknowledges the strong-coupling loophole. Since eq. (4.18) is stated as a general constraint, the authors should either restrict the claim explicitly to power-law histories (with the |n| ≪ 1 case handled by the N_c argument) or add a sentence in §4 stating that non-power-law histories are outside the scope of eq. (4.18). This is a scope qualification, not a rederivation: the n ≥ 1 result eq. (4.12), which covers the physically relevant regime where B grows, is unaffected.

Minor objections (7)

[Section 4, eq. (4.2)] The printed relation P_E = n^2 k^2 η^2 P_B is inverted relative to eqs. (2.5) and (3.8); it should read P_E = n^2/(k^2η^2) P_B. Eq. (4.5) uses the correct relation, so the derivation is unaffected, but the printed eq. (4.2) will confuse readers checking the chain. Relatedly, the intermediate expression |nkη| in eq. (2.5) should be n/|kη|; the final n e^{N_k} is correct.
[Section 4, eq. (4.5)] The numerical factor 1/7 in the p integral should be explained. It corresponds to a specific assumed peak profile (for example, a top-hat of one e-fold above p_B, or P_B = P_B^obs for p ≥ p_B), and since the final coefficient 30 GeV is quoted precisely, the assumed profile should be stated.
[Abstract vs eq. (4.18)] The abstract quotes ρ_inf^{1/4} < 30 GeV (B_obs/10^{-15}G)^{-1} but omits the (p_B/1Mpc^{-1})^{5/4} factor of eq. (4.18). Since p_B ≥ 1 Mpc^{-1}, the abstract quotes the strongest case; this should be stated explicitly.
[Section 4, eq. (4.8)] The inequality N_B ≥ 58.8 - ln(p_B/H_0) + ln(ρ_inf^{1/4}/10^15 GeV) should be explained; it reflects the minimum number of e-folds and the reheating uncertainty, and the reader would benefit from a sentence on which direction is conservative.
[Section 5] The statement that 'a detection of background gravitational waves in the future excludes inflationary magnetogenesis' relies on the single-field slow-roll relation r = 16ε. The paper elsewhere allows the curvature perturbation to be sourced by a spectator field, for which this relation does not hold; the sentence should be qualified to the single-field case.
[Section 3, eq. (3.2)] The approximation δp_nad ≈ (4/3)ρ_em uses ṗ/ρ̇ ≈ -1 during inflation; a one-line derivation would help the reader, since the factor 4/3 is used throughout the subsequent calculation.
[Figure 1] The left panel is a log plot including negative n, which is hard to read; marking the asymptotes f(n) = 1 at n = 1 and n ≈ -0.42 and the divergence as n → 0 would make the figure clearer.

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The paper

The paper claims

If the magnetic fields in cosmic voids were made during inflation, the authors argue, inflation must have run at an energy below about 30 GeV — far below the usual high-energy picture.

The reading

The paper asks whether the magnetic fields inferred in cosmic voids, at least about $10^{-15}$ G on megaparsec scales, could have been manufactured by quantum fluctuations during inflation. The authors' answer is that this works only if inflation ran at a strikingly low energy, $��^{1/4}_{��} < 30$ GeV times a modest dependence on the peak wavenumber of the field and on its present strength. The argument is deliberately model-independent: it uses only the fact that beating the expansion's dilution of the magnetic field requires the vector potential to grow faster than the scale factor squared, which in turn forces the electric field to be far larger than the magnetic field on super-horizon scales. That electric field carries energy, and its energy acts as an isocurvature source for curvature perturbations, which must stay below the amplitude actually seen in the cosmic microwave background. If the bound holds, inflationary magnetogenesis on Mpc scales sits in tension with any high-energy inflationary epoch, and the observed void fields would need an extra amplification after inflation or a dedicated low-scale model.

What carries the argument

The load-bearing mechanism is the electric-field dominance on super-horizon scales. Because the vector potential must grow faster than $a^2$ to outrun adiabatic dilution, the time derivative in $E_i = -a^{-2}�� A_i$ is amplified by the factor $n k ��$, so the electric field exceeds the magnetic field by $n e^{N_k}$. The electric energy then enters the curvature evolution equation $��̇ = -H ��p_{��}/(��+p)$ as an isocurvature pressure, giving $��^{��}_k = 2�� dN ��^{��}_k/(�� _{��})$, which the authors convert into a double time integral over the electric power spectrum $P_E(��,p)$. The power-law ansatz $A_k(��) = (��/��_f)^{-n}A_k(��_f)$ over the last e-fold, together with adiabatic dilution of $P_B$ from the end of inflation to today, converts the observed blazar lower bound on $B$ into a lower bound on the induced curvature power spectrum and hence an upper bound on $��_{��}$.

Extended reading notes

Core claim

The central result is an upper bound on the inflationary energy scale, $��^{1/4}_{��} < 30$ GeV $(p_B/1~��{��}^{-1})^{5/4}(B_{��}/10^{-15}~��)^{-1}$ for peak wavenumbers $p_B �� 1~��{��}^{-1}$, claimed to hold for any power-law growth index $n$ of the vector potential. The chain runs as follows. To explain the observed field without post-inflation amplification, the magnetic field at the end of inflation must be enormous, and since $B �� a^{-2}$ during inflation the vector potential must grow at least as fast as $a^2$, i.e. $A �� |��|^{-n}$ with $n>2$. Equation (2.5) then gives $|E/B| = n k �� = n e^{N_k} �� 1$ on super-horizon scales, where $N_k$ counts e-folds from horizon exit to the end of inflation. The electromagnetic energy density is an isocurvature component, so it sources the curvature perturbation through $��̇ = -H ��p_{��}/(��+p)$ with $��p_{��} �� 4��_{��}/3$, and the induced spectrum is written as a double time integral over the electric power spectrum. Restricting to the last e-fold, taking a power-law vector potential, and requiring the result to stay under the observed $��_�� = 2.2��10^{-9}$ yields the bound. The prefactor $f(n) = [n^2(1-e^{2-2n})/(2n-2)]^{-1/2}$ is at most unity for $|n| �� 1$, so the quoted number is the conservative version; for $|n| �� 1$ the authors argue the bound only tightens, by a factor $��^{1/2}e^{-N_c}$, because the magnetic field must have been much larger at the earlier epoch when $n$ was of order one.

Load-bearing premise

The bound rests on the vector potential being a pure power law in conformal time over the last e-fold of inflation (or over the earlier epoch when its growth index was of order one), and on the magnetic field being diluted only adiabatically between the end of inflation and today.

Editorial extensions

If this is right

If the bound is saturated at $��^{1/4}_{��} �� 10$ GeV, reheating must complete almost immediately, since big bang nucleosynthesis requires a reheating energy above roughly 10 MeV.
The corresponding Hubble scale is $H_{��} < 2��10^{-7}$ eV, so any field responsible for the observed curvature perturbation — the inflaton or a spectator — must be extraordinarily light during inflation and yet decay promptly afterwards.
For single-field slow-roll inflation the same numbers imply a tensor-to-scalar ratio below about $7��10^{-61}$, so a future detection of primordial gravitational waves would rule out an inflationary origin for the void fields.
At such low reheating temperatures, thermal production of dark matter or of a baryon asymmetry is hard, pushing these sectors toward non-thermal mechanisms such as direct inflaton decay.
The constraint can be evaded by any post-inflationary amplification or non-adiabatic dilution of the magnetic field, for instance preheating dynamics or an inverse cascade.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The bound is really a statement about the electric field, not the magnetic field, so any mechanism that generates a large magnetic field while keeping the electric field small — a nearly static vector potential with a large amplitude at horizon crossing — escapes it; the authors themselves flag the strong-coupling regime of the kinetic-coupling model as the place where this could happen, but note
Because the bound scales as $B_{��}^{-1}$, the ongoing improvement in blazar constraints on the void field, and especially the future resolution of the strength–correlation-length degeneracy, directly moves the number: a confirmed field well above $10^{-15}$ G would push the allowed inflationary scale below the electroweak scale.
The same electric-field-as-isocurvature logic should apply to any model that produces a vector field during inflation, including gauged spectator fields and curvaton-like setups, so the constraint is plausibly broader than magnetogenesis proper.
A useful cross-check would be to run the calculation for a vector potential that is not a pure power law over the last e-fold, for example a resonant or oscillatory evolution, to see whether the $f(n) �� 1$ step survives.

Original abstract

Recently, there are several reports that the cosmic magnetic fields on Mpc scale in void region is larger than $\sim 10^{-15}$G with an uncertainty of a few orders from the current blazar observations. On the other hand, in inflationary magnetogenesis models, additional primordial curvature perturbations are inevitably produced from iso-curvature perturbations due to generated electromagnetic fields. We explore such induced curvature perturbations in a model independent way and obtained a severe upper bound for the energy scale of inflation from the observed cosmic magnetic fields and the observed amplitude of the curvature perturbation, as $\rho_{\rm inf}^{1/4} < 30{\rm GeV} \times (B_{\rm obs}/10^{-15}{\rm G})^{-1}$ where $B_{\rm obs}$ is the strength of the magnetic field at present. Therefore, without a dedicated low energy inflation model or an additional amplification of magnetic fields after inflation, inflationary magnetogenesis on Mpc scale is generally incompatible with CMB observations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.41 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.29 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

Recently, there are several reports that the cosmic magnetic fields on Mpc scale in void region is larger than $\sim 10^{-15}$G with an uncertainty of a few orders from the current blazar observations. On the other hand, in inflationary magnetogenesis models, additional primordial curvature perturbations are inevitably produced from iso-curvature perturbations due to generated electromagnetic fields. We explore such induced curvature perturbations in a model independent way and obtained a severe upper bound for the energy scale of inflation from the observed cosmic magnetic fields and the observed amplitude of the curvature perturbation, as $\rho_{\rm inf}^{1/4} < 30{\rm GeV} \times (B_{\rm obs}/10^{-15}{\rm G})^{-1}$ where $B_{\rm obs}$ is the strength of the magnetic field at present. Therefore, without a dedicated low energy inflation model or an additional amplification of magnetic fields after inflation, inflationary magnetogenesis on Mpc scale is generally incompatible with CMB observations.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Recently, there are several reports that the cosmic magnetic fields on Mpc scale in void region is larger than $\sim 10^{-15}$G with an uncertainty of a few orders from the current blazar observations.	background 1.00	0.03	0.02	asserted 0.82
s1	On the other hand, in inflationary magnetogenesis models, additional primordial curvature perturbations are inevitably produced from iso-curvature perturbations due to generated electromagnetic fields.	background 1.00	0.07	0.03	asserted 1.00
s2	We explore such induced curvature perturbations in a model independent way and obtained a severe upper bound for the energy scale of inflation from the observed cosmic magnetic fields and the observed amplitude of the curvature perturbation, as $\rho_{\rm inf}^{1/4} < 30{\rm GeV} \times (B_{\rm obs}/10^{-15}{\rm G})^{-1}$ where $B_{\rm obs}$ is the strength of the magnetic field at present.	result 0.99	0.95	0.73	asserted 1.00
s3	Therefore, without a dedicated low energy inflation model or an additional amplification of magnetic fields after inflation, inflationary magnetogenesis on Mpc scale is generally incompatible with CMB observations.	interpretation 0.99	0.70	0.93	asserted 0.96

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.83.

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The behavior of f(n) defined in eq. (4.11). The left panel is the log plot while the right panel is the log-log plot. It is shown that f(n) = 1 for n = 1 and n ≈ −0.42. One can easily see that f(n) ≤ 1 for |n| ≥ 1 and f(n) ≫ 1 only for |n| ≪ 1. where H −1 0 = 4.4Gpc is the present horizon scale and we have assumed the instantaneous reheating, and then we have P em ζ (k) > e 4×58.8 7 n 2 1 − e 2−2n 2n − 2 2 k pB 3 H0 pB 4 P obs B ργ 2

Figures reproduced from arXiv:1402.0596 by the authors.

References

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View on Pith Pith review arXiv 2013
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M. Giovannini, Entropy perturbations and large-scale magnetic ﬁelds , Class. Quant. Grav. 23, 4991 (2006) [astro-ph/0604134]

View on Pith Pith review arXiv 2006
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M. Giovannini, Tight coupling expansion and fully inhomogeneous magnetic ﬁelds, Phys. Rev. D 74, 063002 (2006) [astro-ph/0606759]

View on Pith Pith review arXiv 2006
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M. Giovannini, Large-scale magnetic ﬁelds, curvature ﬂuctuations and the thermal history of the Universe , Phys. Rev. D 76, 103508 (2007) [arXiv:0707.0857 [astro-ph]]

View on Pith Pith review arXiv 2007
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View on Pith Pith review arXiv 2001
[47]

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View on Pith Pith review arXiv 2011
[48]

R. J. Z. Ferreira, R. K. Jain and M. S. Sloth, “Inﬂationary Magn etogenesis without the Strong Coupling Problem II: Constraints from CMB anisotropies and B-mode s”, arXiv:1403.5516 [astro-ph.CO]

View on Pith Pith review arXiv
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View on Pith Pith review arXiv 2013
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[53]

L. Campanelli, “Evolution of Magnetic Fields in Freely Decaying Magn etohydrodynamic Turbulence,” Phys. Rev. Lett. 98, 251302 (2007) [arXiv:0705.2308 [astro-ph]]

View on Pith Pith review arXiv 2007
[54]

A. Saveliev, K. Jedamzik and G. Sigl, “Evolution of Helical Cosmic Ma gnetic Fields as Predicted by Magnetohydrodynamic Closure Theory,” Phys. Rev. D 87, 123001 (2013) [arXiv:1304.3621 [astro-ph.CO]]

View on Pith Pith review arXiv 2013
[55]

J. Martin and C. Ringeval, “Inﬂation after WMAP3: Confronting the Slow-Roll and Exact Power Spectra to CMB Data,” JCAP 0608, 009 (2006) [astro-ph/0605367]

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J. Martin and C. Ringeval, “First CMB Constraints on the Inﬂatio nary Reheating Temperature,” Phys. Rev. D 82, 023511 (2010) [arXiv:1004.5525 [astro-ph.CO]]. – 16 –

View on Pith Pith review arXiv 2010

Forward citations

Cited by 9 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Gauge-field energy density in inflationary kinetic-coupling models: QFT regularization versus horizon cutoffs
hep-th 2026-10 conditional novelty 6.0 of 10

For a kinetically coupled gauge field during inflation, horizon-cutoff prescriptions for the energy density agree with QFT-regularized results only near |gamma1| tending to 2, and even then only for the total, not for...
Pre-reheating Magnetogenesis in the Kinetic Coupling Model
astro-ph.CO 2016-02 conditional novelty 6.0 of 10

A spectator field that keeps evolving after inflation can amplify magnetic fields enough to explain blazar observations without violating CMB or backreaction constraints.
Large-scale magnetic fields can explain the baryon asymmetry of the Universe
hep-ph 2016-02 conditional novelty 6.0 of 10

Primordial helical hypermagnetic fields, evolving through an inverse cascade before the electroweak transition, can reproduce the observed baryon asymmetry using only standard model physics.
Consistent generation of magnetic fields in axion inflation models
astro-ph.CO 2015-03 conditional novelty 6.0 of 10

Full numerical tracking of axion-inflation magnetogenesis, including post-inflationary production and inverse cascade, yields at most about 10^-19 gauss, below the blazar bound.
Primordial Magnetic Fields from the Post-Inflationary Universe
astro-ph.CO 2014-03 conditional novelty 6.0 of 10

Combining inflationary and post-inflationary (pre-reheating) magnetogenesis produces magnetic fields above 10^-15 G on Mpc scales without strong coupling or electric backreaction, for a demonstrated parameter set.

Show 4 more citing papers

Early Cosmological Evolution of Primordial Electromagnetic Fields
astro-ph.CO 2019-03 conditional novelty 5.5 of 10

Super-horizon magnetic fields in the early universe decay as B squared proportional to a to the minus six times H to the minus two rather than a to the minus four when electric fields dominate, letting high-scale infl...
Light Dark Photon Dark Matter from Inflation
hep-ph 2020-04 conditional novelty 5.0 of 10

Kinetic coupling of a dark photon to the inflaton during inflation yields the observed dark matter abundance for dark photon masses as low as 10^-21 eV, provided the coupling scales roughly as a^-2.
Adding helicity to inflationary magnetogenesis
astro-ph.CO 2014-07 conditional novelty 5.0 of 10

Adding a time-dependent parity-violating coupling to standard inflationary magnetogenesis yields helical magnetic fields that can match intergalactic bounds and seed galactic dynamos at inflation scales of 10^5 to 10^10 GeV.
Inflationary Magnetogenesis without the Strong Coupling Problem II: Constraints from CMB anisotropies and B-modes
astro-ph.CO 2014-03 conditional novelty 4.0 of 10

For the f^2(phi)F^2 inflationary magnetogenesis model, the induced bispectrum gives the strongest constraint, and accepting BICEP2's r = 0.2 caps the present-day field at Mpc scales below 10^-30 G.

How the literature uses it

36 arXiv papers cite this in 55 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States a conflicting result 1 paper

We find that large primordial magnetic fields can be generated during inflation without the strong coupling problem, backreaction problem, or curvature perturbation problem, which seed large-scale magnetic fields with observationally interesting strengths.

Model of inflationary magnetogenesis · 0.81

Papers citing it · 39 sentences without a confident label

Contests it 1 paper

This should be compared with the conservative upper bound obtained for the magnetic field strength today in [ref] [The expression for the minimum amount of inflation appearing in the Planck paper [ref] , also used by the authors of [ref] , carries a misprint [ref] .

Inflationary Magnetogenesis without the Strong Coupling Problem II: Constraints from CMB anisotropies and B-modes · Introduction · contest 0.73

Uses its data or values 1 paper

Combining the corresponding results with the CMB observations, POLARBEAR and NRAO VLA Sky Survey [ref] and the data from the ultra-high-energy cosmic rays [ref] constrain the strength of these fields to be [ref] [equation] The upper and the lower bounds come from the CMB and blazars data, respectively.

Revisiting Magnetogenesis during Inflation · Introduction · 0.93

Compares with it 2 papers

We have found the maximal magnetic fields achievable in the $f^2(ϕ)F_μνF^μν$ model, and compared with the recent constraint derived by Fujita and Yokoyama [ref] .

Inflationary Magnetogenesis without the Strong Coupling Problem II: Constraints from CMB anisotropies and B-modes · Conclusions · 1.00
The magnetic field spectrum in this case is blue-tilted, and similar to [ref] , requires low-scale inflation to have observable magnetic fields on Mpc scales.].

Inflationary magneto-(non)genesis, increasing kinetic couplings, and the strong coupling problem · Introduction · 0.80

Cites it as support 1 paper

Even without this assumption, however, it has been shown in a model-independent manner [ref] that inflationary generation of large-scale magnetic fields is extremely difficult even if the production is limited only to a few last e-folds toward the end of inflation, as long as $n < 0$.

Inflationary magneto-(non)genesis, increasing kinetic couplings, and the strong coupling problem · Constraints · 0.97

Cites it as background 8 papers

The back reaction problem by the generation of electromagnetic fields during inflation has been found [ref] (for more recent related works on the relation between the generated gauge fields and inflation, see [ref] ).

Generation of large-scale magnetic fields, non-Gaussianity, and primordial gravitational waves in inflationary cosmology · Issues of the backreaction and the strong coupling · 0.84
Much of the work has assumed that either the coupling function $f$ [ref] or some quantities more directly related to the magnetic field [ref] is a power law, though [ref] considered a more involved form.

Narrowing the window of inflationary magnetogenesis · Introduction · 0.95
A large body of work has explored the possibility that magnetogenesis can be incorporated into the context of inflationary cosmology [ref] .

Inflationary magneto-(non)genesis, increasing kinetic couplings, and the strong coupling problem · Introduction · 1.00

What it cites · 30 sentences without a confident label

Uses data or values from 5 papers

where $H_0^-1 = 4.4$Gpc is the present horizon scale and we have assumed the instantaneous reheating, and then we have [equation] Finally, by requiring that the induced curvature perturbations given by the above expression should not exceed the observed power spectrum $_ζ^(k) =2.2×10^-9$ at the Planck pivot scale $k^-1=20Mpc$ [ref] , we can obtain the upper bound on the inflationary energy scale as

Planck 2013 results. XXII. Constraints on inflation · Model Independent Constraint · 0.99
Such void magnetic fields could be detected by blazar observations [ref] and it is reported that their strength is larger than $∼10^-15$G with an uncertainty of a few orders.

Constraints on the Intergalactic Magnetic Field from Gamma-Ray Observations of Blazars · Introduction · 0.81
Such void magnetic fields could be detected by blazar observations [ref] and it is reported that their strength is larger than $∼10^-15$G with an uncertainty of a few orders.

Determination of intergalactic magnetic fields from gamma ray data · Introduction · 0.81

Cites as support 1 paper

But no mechanism that leads to such cancellation is found [ref] .]

Universal upper limit on inflation energy scale from cosmic magnetic field · Power spectrum of Induced Curvature Perturbations · 0.98

Cites as background 16 papers

Although many models of the generation of magnetic fields during inflation are proposed so far [ref] , it is known that these inflationary magnetogenesis models suffer from several problems, namely the strong coupling problem [ref] , the backreaction problem [ref] , the anisotropy problem [ref] and the curvature perturbation problem [ref] .

Inflationary Magnetogenesis without the Strong Coupling Problem · Introduction · 0.88
Although many models of the generation of magnetic fields during inflation are proposed so far [ref] , it is known that these inflationary magnetogenesis models suffer from several problems, namely the strong coupling problem [ref] , the backreaction problem [ref] , the anisotropy problem [ref] and the curvature perturbation problem [ref] .

Higher order statistics of curvature perturbations in IFF model and its Planck constraints · Introduction · 0.88
Although many models of the generation of magnetic fields during inflation are proposed so far [ref] , it is known that these inflationary magnetogenesis models suffer from several problems, namely the strong coupling problem [ref] , the backreaction problem [ref] , the anisotropy problem [ref] and the curvature perturbation problem [ref] .

Magnetic fields from inflation? · Introduction · 0.88

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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